Enumeration of -strata in quantum matrices with respect to dimension

We present a combinatorial method to determine the dimension of H -strata in the algebra of m × n quantum matrices O q ( M m , n ( K ) ) as follows. To a given H -stratum we associate a certain permutation via the notion of pipe dreams. We show that the dimension of the H -stratum is precisely the number of odd cycles in this permutation. Using this result, we are able to give closed formulas for the trivariate generating function that counts the d-dimensional H -strata in O q ( M m , n ( K ) ) . Finally, we extract the coefficients of this generating function in order to settle conjectures proposed by the first and third named authors (Bell and Launois (2010) [3], Bell, Launois and Lutley (2010) [4]) regarding the asymptotic proportion of d-dimensional H -strata in O q ( M m , n ( K ) ) .